# Integral coefficient for internal energy of ionic liquid

#### Geraili

I have a question, and I will be really grateful if someone helps me. I have a polynomial equation for internal energy which I calculated by integration an equation of state formula, which is based on density. But, because I calculated this using integration one integration constant which is temperature dependent( based on other articles) that I don't know how can I formulate it to have its magnitude to calculate internal energy at other range of data. My simulation box has contained 200 molecules of ionic liquid with one negative ion( PF6) and a positive one( butyl methyl imidazolium). Because according to internal energy equation at zero density internal energy is equal to the integration constant, we considered it as ionic liquid internal energy at ideal gas state. With all those in mind, how can I use a degree of freedom of rotational, vibrational, and translational to formulate this integration constant dependent of temperature to use it in other range of data? Or, is there any other method to formulate it? Thanks very much in advance

#### Woody

Without knowing the fine details of your problem
the only thought I can suggest is to follow your equation to the limits
and see if this can at least bound your solution.

You say your equation is based on density,
can we make assumptions about the solution as the density tends toward zero?

Can you go back to the original equation and define some limits in the original equation which will guide the constant?

Is the original formula based on a polynomial curve fit to experimental data,
or on an analytical description of the underlying physics?

#### Geraili

the main equation

I am doing molecular dynamics simulation. Internal energy in my system means total energy because it is thermodynamics system. I need integration constant because I want to report the thermodynamic properties, not their change. I want to explain it more in detail, so you can help me.
Ein=∫Pin/rho^2 d rho +F(T)= RT{ (e*rho^2)+(f*rho)+(g*rho^4)}+F(T)
Cp=R{ (a*rho^2)+(b*rho)+(c*rho^4)}+F'(T)
in these formulas: rho= density. e,f,g,a,b,c = temperature dependant coefficients. Pin= internal pressure. Ein= internal energy=total energy. F(T)= integration constant. F'(T)= temperature derivation of integration constant.

I calculate enthalpy from Ein and then calculate Cp. My main problem is that according to the article that I used, these integration constants, which are y-intercept,( F(T) and F'(T)) should be an ideal contribution of corresponding thermodynamical properties, but in MD and its numerical solutions, these parameters are just y-intercepts. So if I have enough data in MD (from high rho to zero rho) these y-intercepts would be an ideal contribution( because they are in rho=0) and it would be easy for me to have them. In my case, there is no way to calculate all the rho until zero( I work between 4.8-4.9 mol/lit), so I just extrapolate the thermodynamical properties against rho and then find y-intercept which is not ideal contribution ( and not temperature dependent and fluctuational to temperature) because this charts cannot properly extrapolate and find the exact ideal contribution( the thermodynamical properties at zero rho) because every extrapolation in different temperature would find the easiest way to reach y-intercept. So I cannot say that this is an ideal contribution at all, I need these y-intercepts( in my case not ideal contribution) to calculate the properties correctly, but the only way could be some mathematical methods to calculate F(T) and F'(T). Do you know any methods? or is there anything which you can advice me to do?

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#### Geraili

the reference article

I think it is a little confusing, so I uploaded the article that I used for mine calculations. As you can see in this article, the author used "3/2 RT" for F(T) because it was monoatomic gas, so in my case, I should consider the F(T) equal to 3RT( it is an ionic liquid fluid which is polyatomic)( right?????). But the problem is in my case I don't have the data to zero density like the author of this article had. I mean, his F(T) which is y-intercept accurately became ideal contribution (3/2 RT) because he traced the data until zero density and y-intercept matched the exact amount of F(T). In my case, I used Molecular Dynamics Simulation and because of force field limitation, I cannot have the data to zero density. So consequently, the F(T) in my formula would become a y-intercept which is not equal to really ideal contribution or 3RT because the data( density= 4-5 mol/lit) would be extrapolated to find their corresponding y-intercept, which ideally should be ideal contribution, but here, they are ideal contribution plus an error for extrapolation ( or not having the exact data to zero density). I know, I am a little confused because of so much overanalyzing.
If you were in my place, and in the above equation for Cp you had everything but F'(T) how would you calculate that( considering the article)??
thank you very much for your time and consideration.

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